## Kyoto Journal of Mathematics

### Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$-elementary lattices

#### Abstract

We give a classification of all nonsymplectic automorphisms of prime order $p$ acting on irreducible holomorphic symplectic four-folds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for $p=2,3$, and $7\leq p\leq19$. Our classification relates some invariants of the fixed locus to the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients. In several cases we provide explicit examples. As an application, we find new examples of nonnatural nonsymplectic automorphisms of order $3$.

#### Article information

Source
Kyoto J. Math., Volume 56, Number 3 (2016), 465-499.

Dates
Revised: 22 April 2015
Accepted: 23 April 2015
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.kjm/1471872277

Digital Object Identifier
doi:10.1215/21562261-3600139

Mathematical Reviews number (MathSciNet)
MR3542771

Zentralblatt MATH identifier
1375.14143

#### Citation

Boissière, Samuel; Camere, Chiara; Sarti, Alessandra. Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$ -elementary lattices. Kyoto J. Math. 56 (2016), no. 3, 465--499. doi:10.1215/21562261-3600139. https://projecteuclid.org/euclid.kjm/1471872277

#### References

• [1] M. Artebani and A. Sarti, Non-symplectic automorphisms of order $3$ on K3 surfaces, Math. Ann. 342 (2008), 903–921.
• [2] M. Artebani, A. Sarti, and S. Taki, K3 surfaces with non-symplectic automorphisms of prime order, with an appendix by S. Kondō, Math. Z. 268 (2011), 507–533.
• [3] A. Beauville, “Some remarks on Kähler manifolds with $c_{1}=0$” in Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progr. Math. 39, Birkhäuser, Boston, 1983, 1–26.
• [4] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), 755–782.
• [5] A. Beauville, Antisymplectic involutions of holomorphic symplectic manifolds, J. Topol. 4 (2011), 300–304.
• [6] A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 703–706.
• [7] S. Boissière, Automorphismes naturels de l’espace de Douady de points sur une surface, Canad. J. Math. 64 (2012), 3–23.
• [8] S. Boissière, C. Camere, G. Mongardi, and A. Sarti, Isometries of ideal lattices and hyperkähler manifolds, Int. Math. Res. Notes (IMRN) (2016), 963–977.
• [9] S. Boissière, M. Nieper-Wisskirchen, and A. Sarti, Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties, J. Math. Pures Appl. (9) 95 (2011), 553–563.
• [10] S. Boissière, M. Nieper-Wisskirchen, and A. Sarti, Smith theory and irreducible holomorphic symplectic manifolds, J. Topol. 6 (2013), 361–390.
• [11] S. Boissière and A. Sarti, A note on automorphisms and birational transformations of holomorphic symplectic manifolds, Proc. Amer. Math. Soc. 140 (2012), 4053–4062.
• [12] C. Camere, Symplectic involutions of holomorphic symplectic four-folds, Bull. Lond. Math. Soc. 44 (2012), 687–702.
• [13] C. Camere, Lattice polarized irreducible holomorphic symplectic manifolds, Ann. Inst. Fourier 66 (2016), 687–709.
• [14] C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.
• [15] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third ed., Grundlehren Math. Wiss. 290, Springer, New York, 1999.
• [16] I. V. Dolgachev, Integral quadratic forms: applications to algebraic geometry (after V. Nikulin), Astérisque 105, Soc. Math. France, Paris, 1983, 251–278, Séminaire Bourbaki 1982/1983, no. 611.
• [17] I. V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci. 81 (1996), 2599–2630.
• [18] I. V. Dolgachev and S. Kondō, “Moduli of K3 surfaces and complex ball quotients” in Arithmetic and Geometry around Hypergeometric Functions, Progr. Math. 260, Birkhäuser, Basel, 2007, 43–100.
• [19] I. V. Dolgachev, B. van Geemen, and S. Kondō, A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces, J. Reine Angew. Math. 588 (2005), 99–148.
• [20] V. González-Aguilera and A. Liendo, Automorphisms of prime order of smooth cubic $n$-folds, Arch. Math. (Basel) 97 (2011), 25–37.
• [21] E. Horikawa, On deformations of holomorphic maps, III, Math. Ann. 222 (1976), 275–282.
• [22] D. Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), 63–113.
• [23] D. Huybrechts, A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky], Astérisque 348 (2012), 375–403, Séminaire Bourbaki 2010/2011, no. 1040.
• [24] M. Joumaah, Non-symplectic involutions of irreducible symplectic manifolds of $\mathit{K3}^{[n]}$-type, Math. Z. online 20 January 2016.
• [25] S. Kondō, Automorphisms of algebraic K3 surfaces which act trivially on Picard groups, J. Math. Soc. Japan 44 (1992), 75–98.
• [26] E. Markman, “A survey of Torelli and monodromy results for holomorphic-symplectic varieties” in Complex and Differential Geometry, Springer Proc. Math. 8, Springer, Heidelberg, 2011, 257–322.
• [27] H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963/1964), 347–361.
• [28] G. Mongardi, Symplectic involutions on deformations of K3$^{[2]}$, Cent. Eur. J. Math. 10 (2012), 1472–1485.
• [29] G. Mongardi, Automorphisms of hyperkähler manifolds, Ph.D. dissertation, Universitá degli Studi di Roma 3, Rome, preprint, arXiv:1303.4670v1 [math.AG].
• [30] D. R. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.
• [31] Y. Namikawa, Periods of Enriques surfaces, Math. Ann. 270 (1985), 201–222.
• [32] V. V. Nikulin, Integer symmetric bilinear forms and some of their applications (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238; English translation in Math. USSR Izv. 14 (1980), 103–167.
• [33] V. V. Nikulin, Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by $2$-reflections (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 109–188; English translation in J. Soviet Math. 22 (1983), 1401–1475.
• [34] K. G. O’Grady, Irreducible symplectic $4$-folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134 (2006), 99–137.
• [35] K. Oguiso and S. Schröer, Enriques manifolds, J. Reine Angew. Math. 661 (2011), 215–235.
• [36] H. Ohashi and M. Wandel, Non-natural non-symplectic involutions on symplectic manifolds of K3$^{[2]}$-type, preprint, arXiv:1305.6353v2 [math.AG].
• [37] A. N. Rudakov and I. R. Shafarevich, “Surfaces of type K3 over fields of finite characteristic” in Current Problems in Mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, 115–207.
• [38] S. Taki, Non-symplectic automorphisms of $3$-power order on K3 surfaces, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 125–130.
• [39] M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996), 601–611.
• [40] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spéc. 10, Soc. Math. France, Paris, 2002.